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### Disjunctive programming and relaxations of polyhedra

Conforti, Michele
Del Pia, Alberto
Mathematical Programming. - 2014/144/1-2/307-314
Disponible
Titre: Disjunctive programming and relaxations of polyhedra
Auteur: Conforti, Michele
Contributeur: Del Pia, Alberto
Sujet: Mixed integer programming - Disjunctive programming - Polyhedral relaxations
Description: Given a polyhedron $$L$$ with $$h$$ facets, whose interior contains no integral points, and a polyhedron $$P$$ , recent work in integer programming has focused on characterizing the convex hull of $$P$$ minus the interior of $$L$$ . We show that to obtain such a characterization it suffices to consider all relaxations of $$P$$ defined by at most $$n(h-1)$$ among the inequalities defining $$P$$ . This extends a result by Andersen, Cornuéjols, and Li.
Publication en relation: Mathematical Programming. - 2014/144/1-2/307-314
Document hôte: Mathematical Programming
Identifiant: 10.1007/s10107-013-0634-3 (DOI)
• Plusieurs versions

### A geometric approach to cut-generating functions

Basu, Amitabh, Conforti, Michele, Di Summa, Marco
Mathematical Programming, 2015, Vol.151(1), pp.153-189 [Revue évaluée par les pairs]

• Plusieurs versions

### The projected faces property and polyhedral relations

Conforti, Michele, Pashkovich, Kanstantsin
Mathematical Programming, 2016, Vol.156(1), pp.331-342 [Revue évaluée par les pairs]

• Article
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### Maximal $S$-Free Convex Sets and the Helly Number

Conforti, Michele, Summa, Marco Di
SIAM Journal on Discrete Mathematics, 2016, Vol.30(4), pp.2206-2216 [Revue évaluée par les pairs]
SIAM Journals (Society for Industrial and Applied Mathematics), Copyright �� by the Society of Industrial and Applied Mathematics, Philadelphia, PA
Disponible
Titre: Maximal $S$-Free Convex Sets and the Helly Number
Auteur: Conforti, Michele; Summa, Marco Di
Sujet: $S$-Free Convex Sets ; Helly Number ; Cut-Generating Functions ; 52c07 ; 90c10
Description: Given a subset $S$ of $\mathbb{R}^d$, the Helly number $h(S)$ is the largest size of an inclusionwise minimal family of convex sets whose intersection is disjoint from $S$. A convex set is $S$-free if its interior contains no point of $S$. The parameter $f(S)$ is the largest number of maximal faces in an inclusionwise maximal $S$-free convex set. We study the relation between the parameters $h(S)$ and $f(S)$. Our main result is that $h(S)\le (d+1)f(S)$ for every nonempty proper closed subset $S$ of $\mathbb{R}^d$. We also study the Helly number of the Cartesian product of two discrete sets.
Fait partie de: SIAM Journal on Discrete Mathematics, 2016, Vol.30(4), pp.2206-2216
Identifiant: 0895-4801 (ISSN); 1095-7146 (E-ISSN); 10.1137/16M1063484 (DOI)
• Plusieurs versions

### Disjunctive programming and relaxations of polyhedra

Conforti, Michele, Del Pia, Alberto
Mathematical Programming, 2014, Vol.144(1), pp.307-314 [Revue évaluée par les pairs]

• Article
Sélectionner

### The Projected Faces Property and Polyhedral Relations

Conforti, Michele, Pashkovich, Kanstantsin
Cornell University
Disponible
Titre: The Projected Faces Property and Polyhedral Relations
Auteur: Conforti, Michele; Pashkovich, Kanstantsin
Sujet: Mathematics - Combinatorics
Description: Margot (1994) in his doctoral dissertation studied extended formulations of combinatorial polytopes that arise from "smaller" polytopes via some composition rule. He introduced the "projected faces property" of a polytope and showed that this property suffices to iteratively build extended formulations of composed polytopes. For the composed polytopes, we show that an extended formulation of the type studied in this paper is always possible only if the smaller polytopes have the projected faces property. Therefore, this produces a characterization of the projected faces property. Affinely generated polyhedral relations were introduced by Kaibel and Pashkovich (2011) to construct extended formulations for the convex hull of the images of a point under the action of some finite group of reflections. In this paper we prove that the projected faces property and affinely generated polyhedral relation are equivalent conditions.
Identifiant: 1305.3782 (ARXIV ID)
• Article
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### The Projected Faces Property and Polyhedral Relations

Conforti, Michele, Pashkovich, Kanstantsin
arXiv.org, Oct 10, 2014
© ProQuest LLC All rights reserved, Engineering Database, Publicly Available Content Database, ProQuest Engineering Collection, ProQuest Technology Collection, ProQuest SciTech Collection, Materials Science & Engineering Database, ProQuest Central (new), ProQuest Central Korea, SciTech Premium Collection, Technology Collection, ProQuest Central Essentials, ProQuest One Academic, Engineering Collection (ProQuest)
Disponible
Titre: The Projected Faces Property and Polyhedral Relations
Auteur: Conforti, Michele; Pashkovich, Kanstantsin
Contributeur: Pashkovich, Kanstantsin (pacrepositoryorg)
Sujet: Hulls (Structures) ; Polytopes ; Combinatorial Analysis ; Convexity ; Formulations
Description: Margot (1994) in his doctoral dissertation studied extended formulations of combinatorial polytopes that arise from "smaller" polytopes via some composition rule. He introduced the "projected faces property" of a polytope and showed that this property suffices to iteratively build extended formulations of composed polytopes. For the composed polytopes, we show that an extended formulation of the type studied in this paper is always possible only if the smaller polytopes have the projected faces property. Therefore, this produces a characterization of the projected faces property. Affinely generated polyhedral relations were introduced by Kaibel and Pashkovich (2011) to construct extended formulations for the convex hull of the images of a point under the action of some finite group of reflections. In this paper we prove that the projected faces property and affinely generated polyhedral relation are equivalent conditions.
Fait partie de: arXiv.org, Oct 10, 2014
Identifiant: 2331-8422 (E-ISSN)
• Article
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### An extreme function which is nonnegative and discontinuous everywhere

Basu, Amitabh, Conforti, Michele
arXiv.org, Feb 5, 2018
© ProQuest LLC All rights reserved, Engineering Database, Publicly Available Content Database, ProQuest Engineering Collection, ProQuest Technology Collection, ProQuest SciTech Collection, Materials Science & Engineering Database, ProQuest Central (new), ProQuest Central Korea, SciTech Premium Collection, Technology Collection, ProQuest Central Essentials, ProQuest One Academic, Engineering Collection (ProQuest)
Disponible
Titre: An extreme function which is nonnegative and discontinuous everywhere
Auteur: Basu, Amitabh; Conforti, Michele
Contributeur: Conforti, Michele (pacrepositoryorg)
Sujet: Mathematical Models ; Optimization and Control ; Functional Analysis
Description: We consider Gomory and Johnson's infinite group model with a single row. Valid inequalities for this model are expressed by valid functions and it has been recently shown that any valid function is dominated by some nonnegative valid function, modulo the affine hull of the model. Within the set of nonnegative valid functions, extreme functions are the ones that cannot be expressed as convex combinations of two distinct valid functions. In this paper we construct an extreme function $$\pi:\mathbb{R} \to [0,1]$$ whose graph is dense in $$\mathbb{R} \times [0,1]$$. Therefore, $$\pi$$ is discontinuous everywhere.
Fait partie de: arXiv.org, Feb 5, 2018
Identifiant: 2331-8422 (E-ISSN)
• Article
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### Balas formulation for the union of polytopes is optimal

Conforti, Michele, Faenza, Yuri
arXiv.org, Nov 2, 2017
© ProQuest LLC All rights reserved, Engineering Database, Publicly Available Content Database, ProQuest Engineering Collection, ProQuest Technology Collection, ProQuest SciTech Collection, Materials Science & Engineering Database, ProQuest Central (new), ProQuest Central Korea, SciTech Premium Collection, Technology Collection, ProQuest Central Essentials, ProQuest One Academic, Engineering Collection (ProQuest)
Disponible
Titre: Balas formulation for the union of polytopes is optimal
Auteur: Conforti, Michele; Faenza, Yuri
Contributeur: Faenza, Yuri (pacrepositoryorg)
Sujet: Hulls (Structures) ; Polytopes ; Inequalities ; Convexity ; Polynomials
Description: A celebrated theorem of Balas gives a linear mixed-integer formulation for the union of two nonempty polytopes whose relaxation gives the convex hull of this union. The number of inequalities in Balas formulation is linear in the number of inequalities that describe the two polytopes and the number of variables is doubled. In this paper we show that this is best possible: in every dimension there exist two nonempty polytopes such that if a formulation for the convex hull of their union has a number of inequalities that is polynomial in the number of inequalities that describe the two polytopes, then the number of additional variables is at least linear in the dimension of the polytopes. We then show that this result essentially carries over if one wants to approximate the convex hull of the union of two polytopes and also in the more restrictive setting of lift-and-project.
Fait partie de: arXiv.org, Nov 2, 2017
Identifiant: 2331-8422 (E-ISSN)
• Article
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### Reverse Split Rank

Conforti, Michele, Faenza, Yuri
arXiv.org, Oct 12, 2014
© ProQuest LLC All rights reserved, Engineering Database, Publicly Available Content Database, ProQuest Engineering Collection, ProQuest Technology Collection, ProQuest SciTech Collection, Materials Science & Engineering Database, ProQuest Central (new), ProQuest Central Korea, SciTech Premium Collection, Technology Collection, ProQuest Central Essentials, ProQuest One Academic, Engineering Collection (ProQuest)
Disponible
Titre: Reverse Split Rank
Auteur: Conforti, Michele; Faenza, Yuri
Contributeur: Faenza, Yuri (pacrepositoryorg)
Sujet: Stock Splits ; Stock Market Delistings ; Polyhedra ; Polyhedrons ; Integrals ; Optimization and Control ; Discrete Mathematics
Description: The reverse split rank of an integral polyhedron P is defined as the supremum of the split ranks of all rational polyhedra whose integer hull is P. Already in R^3 there exist polyhedra with infinite reverse split rank. We give a geometric characterization of the integral polyhedra in R^n with infinite reverse split rank.
Fait partie de: arXiv.org, Oct 12, 2014
Identifiant: 2331-8422 (E-ISSN)